Axiomatic systems cannot be used to justify themselves. This has been empirically demonstrated with the quantitative system of mathematical number theory. The result of axiomatic justification using the axioms themselves bears an asymptotic relationship to the justification we wish to reach. That is, by infinitely increased semantic analysis we can arrive infinitely closer to justifying the axioms but never actually intersect with the justification we approach.
As the number theory results reflect, the most accurate conclusion possible in justifying axioms using themselves is proof that no conclusion is possible. Otherwise, we can speculate as to the source of the justification we seek, without ever actually knowing it. Specifically, since we know that the subscribed subjects of axiomatic systems can be justified by such systems, and these subjects may represent embedded axioms for their subjects, it can be inferred that our more fundamental axioms may be the embedded subjects of a greater axiomatic system that is then, inherently, transcendent.
Our asymptotically semantic analysis of our axioms using themselves was perhaps advanced the most by Hume, who implied that the fundamental axioms of our thought are the embedded subjects of our nervous system. To further the asymptotic analysis, we may further explore this relationship or the properties of the nervous system itself. Regardless of either pursuit, however, is the certainty that the objective such analysis seeks can inherently never be reached.
